How many ways are there to arrange the letters a, b, c, d and e such that b does

Question

How many ways are there to arrange the letters a, b, c, d and e such that b does not immediately follow a? Suppose the students in a class are all either freshmen, sophomores, or juniors. How many students must be enrolled in the class to guarantee that there are at least 10 students in the same grade level? Suppose that there are 21 students in a class, and each student is either a freshman, sophomore, or junior. Show that there must be at least 6 freshmen or at least 10 sophomore or at least 7 juniors.

Explanation / Answer

1.
Answer: 96
Explanation: a,b,c,d,e = 5 variables, so total combinations = 5!= 120
if [ab] joined, then the variables would be [ab],c,d,e= 4 variables, so total combinations to avoid is 4!= 24
Therefore number of combinations which are legible are 5!-4!= 96;

b.
Answer 10
Explanation: given that all the students are either freshmen or sophomore or junior. But it is not necessarily to be at least 1 student in each group. So if there are only 10 students, then everyone can belong to same grade level.

c.
If there are less than or equal 5 freshmen, less than or equal 9 sophomores, and less than or equal 6 juniors in the class, then altogether there are no more than 20. students in the class, which is not the case. Therefore, our assumption is wrong, and there are either at least 6 freshmen, at least 10 sophomores, or at least 7 juniors in the class.

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